Formulas and Notes for Statistical Inference: Confidence Intervals and Significance Tests
Type of Data
Standard Deviation (SD)
Confidence Interval
∗
𝑝̂ ± 𝑧 𝑆𝐷
Proportion
2 Sample
Difference of
Proportions
 Be sure to label
p1 and p2
𝑝(1 − 𝑝)
√
𝑛
𝑝1 (1 − 𝑝1 ) 𝑝2 (1 − 𝑝2 )
√
+
𝑛1
𝑛2
OR
1
1
√𝑝(1 − 𝑝)√ +
𝑛1 𝑛2
for pooled when you assume p1=p2. P is the
total proportion for the pooled results.
Mean
𝑠
df=n-1
√𝑛
2 Sample
Difference of
Means
df=minimum n-1
Or calculator
𝑠12
𝑠22
√ +
𝑛1 𝑛2
Use the 𝑝̂ in the SD
Calc: 1-PropZInt
(𝑝̂ 1 - 𝑝̂ 2)± 𝑧 ∗ 𝑆𝐷
You would typically use the 1st
SD expression as we don’t
know that p1=p2.
Calc: 2-PropZInt
𝑥̅ ± 𝑡 ∗ 𝑆𝐷
Note that you are using the t
distribution because we don’t
know σ.
Calc: TInterval
Significance Test Ho , Ha
Ho : po=claim
Ha : po ≠, <, > claim
Use po in the SD
Calc: Z-Test
Ho : p1= p2
Ha : p1 ≠, <, > p2
We typically use the pooled SD
expression when finding the Ztest as we are assuming p1=p2.
Calc: 2-PropZTest
Ho : µo=claim
Ha : µo ≠, <, > claim
Calc: T-Test
(𝑥̅1 − 𝑥̅2 ) ± 𝑡 ∗ 𝑆𝐷
Ho : µ1 = µ2
Ha : µ1 ≠, <, > µ2
Calc: 2-SampTInt
Calc: 2-SampTTest
Be sure labe x1 and x2
𝑑̅ ± 𝑡 ∗ 𝑆𝐷
Paired Comparison
of Differences
𝑠𝑑
df= n-1
√𝑛
To be used when each
sample set is not
independent of the other:
“before /after”
Where 𝑑̅ is the mean difference.
Be sure to show what order you
are subtracting.
Where sd is the SD of the differences between each.
Calc: TInterval (Make sure to
have a list of the differences)
Ho : µd = 0
Ha : µd ≠, <, > 0
Calc: T-Test

A 95% confidence interval can be used as an equivalent method - compared to a 2-tailed 5% level of significance test for a difference.

Identify the type of problem first, then determine the appropriate SD formula to use, use this in either the Confidence Interval formula or in
finding the z or t – test statistic.

Use a z-test stat / critical value for proportions. Use the t-distribution for the means of a sample set when you don’t know the population
SD, σ.
o Note: If you do know the population SD, σ, then you can use the z-test stat / critical value even for a mean problem.

Be sure to always check conditions to start an inference problem.

Be sure to label your populations being used in the problem.

Be sure to follow the steps for a significance test, and show these 4 steps specifically.

Draw the normal distribution curve with the values being used.

State your final conclusions using the statistical reasoning and in context of the problem.

A confidence interval of differences that is all negative or all positive is statistical evidence to suggest an actual difference.
o

This would be the same as a 2-Tailed significance test where we reject the null hypothesis.
A confidence interval of differences that extends from negative to positive does not show statistical evidence of a difference.
o
This would be the same as a 2-Tailed significance test where we do not reject the null hypothesis.